Let's move beyond three-sided shapes and add another side. A quadrilateral is any polygon with four sides, four vertices (corners), and four angles. It's a broad category that covers a lot of familiar shapes.

Just like triangles, quadrilaterals come in many forms. We classify them based on their properties, like the length of their sides, the size of their angles, and whether any sides are parallel.

One of the most important groups of quadrilaterals is the parallelogram family. These shapes all share a fundamental property: they have two pairs of parallel sides. This simple rule leads to a whole set of useful characteristics.

Let's break down each member of this family.

Because its opposite sides are parallel, a parallelogram has a few other traits:

• Opposite sides are equal in length.
• Opposite angles are equal.
• Diagonals bisect each other (they cross at their midpoints).

Since a rectangle is a type of parallelogram, it has all the properties of one. But its right angles give it an extra feature: its diagonals are equal in length.

A rhombus is also a parallelogram, so it shares those core properties. Its unique features come from its equal sides:

• The diagonals are perpendicular; they intersect at a 90° angle.
• The diagonals also bisect the corner angles (cut them in half).

A square is the most specific type of parallelogram. It's a rectangle (because it has four right angles) and a rhombus (because it has four equal sides). This means it has all the properties of parallelograms, rectangles, and rhombuses combined.

It can be helpful to see the properties side-by-side. Notice how each shape inherits the properties of the one before it, then adds something new.

Understanding these basic four-sided shapes is a crucial first step. Their definitions and properties are the building blocks for exploring more complex geometric figures and relationships.